We introduce the fundamental group ${\mathcal F}(A)$ of
a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple)
densely defined lower semicontinuous trace.
This is a generalization of ``Fundamental Group of Simple
$C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani.
Our definition in this paper makes sense for stably projectionless $C^*$-algebras.
We show that there exist separable stably projectionless $C^*$-algebras such that
their fundamental groups are equal to $\mathbb{R}_+^\times$
by using the classification theorem of Razak and Tsang.
This is a contrast to the unital case in Nawata and Watatani.
This study is motivated by the work of Kishimoto and Kumjian.

Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit
of a sequence
$$
A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3}
\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots,
$$
where
$A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$,
$X^{i}_n$ are $[0,1]$, $k_n$, and
$[n,i]$ are positive integers.
Suppose that $A$ has the
ideal property: each closed two-sided ideal of $A$ is generated by
the projections inside the ideal, as a closed two-sided ideal.
In this article, we give a complete classification of AI algebras with the ideal property.

For monotone complete $C^*$-algebras
$A\subset B$ with $A$ contained in $B$ as a monotone closed
$C^*$-subalgebra, the relation $X = AsA$
gives a bijection between the set of all
monotone closed linear subspaces $X$ of $B$ such that
$AX + XA \subset X$
and
$XX^* + X^*X \subset A$
and a set of certain partial
isometries $s$ in the ``normalizer" of $A$ in $B$,
and similarly for the map $s \mapsto \Ad s$
between the latter set and a set of certain ``partial $*$-automorphisms"
of $A$.
We introduce natural inverse semigroup
structures in the set of such $X$'s and the set of
partial $*$-automorphisms of $A$, modulo a certain relation, so that
the composition of these maps induces an inverse semigroup
homomorphism between them.
For a large enough $B$ the homomorphism becomes surjective and
all the partial $*$-automorphisms of
$A$ are realized via partial isometries in $B$.
In particular, the inverse semigroup associated with
a type ${\rm II}_1$ von Neumann factor,
modulo the outer automorphism group,
can be viewed as the fundamental group of the factor.
We also consider the $C^*$-algebra version of these results.

We study the type decomposition and the rectangular AFD property for
$W^*$-TRO's. Like von Neumann algebras, every $W^*$-TRO can be
uniquely decomposed into the direct sum of $W^*$-TRO's of
type $I$, type $II$, and type $III$.
We may further consider $W^*$-TRO's of type $I_{m, n}$
with cardinal numbers $m$ and $n$, and consider $W^*$-TRO's of
type $II_{\lambda, \mu}$ with $\lambda, \mu = 1$ or $\infty$.
It is shown that every separable stable $W^*$-TRO
(which includes type $I_{\infty,\infty}$, type $II_{\infty,
\infty}$ and type $III$) is TRO-isomorphic to a von Neumann algebra.
We also introduce the rectangular version of the approximately finite
dimensional property for $W^*$-TRO's.
One of our major results is to show that a separable $W^*$-TRO
is injective if and only
if it is rectangularly approximately finite dimensional.
As a consequence of this result, we show that a dual operator space
is injective if and only if its operator predual is a rigid
rectangular ${\OL}_{1, 1^+}$ space (equivalently, a rectangular